Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Abstract

Let\(\mathfrak{X}\) be a barreled locally convex space. A continuous operator Ξ on\(\mathfrak{X}\) is called anequicontinuous generator if {Ξn/n!;n=0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series. There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators Ξ1, Ξ2 satisfy [Ξ1,Ξ2]=αΞ2 for some α∈ℂ thenaΞ1+bΞ2 is also an equicontinuous generator for anya, b∈ℂ. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.